Factorising and Identities: Quick Revision

A brief note on Factorising and Identities. Read it before a test. Try each question at the end before you open its answer.

Factorising

• A common factor first. $6x^2 - 9x = 3x(2x - 3)$.
• A quadratic $ax^2 + bx + c$. Find two numbers that multiply to $ac$ and add up to $b$. Split $bx$ into them and group.
• A cubic. Test the divisors of the constant term. When $p(a) = 0$, divide by $x - a$ and factorise the quadratic left over.

The identities

A worked example

Factorise $x^3 - 3x^2 - 10x + 24$.

1. Test. $p(2) = 8 - 12 - 20 + 24 = 0$, so $x - 2$ is a factor.
2. Divide. $x^3 - 3x^2 - 10x + 24 = (x - 2)(x^2 - x - 12)$.
3. The quadratic. $x^2 - x - 12 = (x - 4)(x + 3)$.

So it is $(x - 2)(x - 4)(x + 3)$.

Practice with new numbers

A program to try

The program finds the zeroes of the worked example by testing.

Mistakes to avoid

• Forgetting the middle term. $(x + 5)^2 = x^2 + 10x + 25$, not $x^2 + 25$.
• Wrong signs in a sum of cubes. $x^3 + y^3$ has $-xy$ in its second factor.
• Testing numbers that cannot work. Only divisors of the constant term can be whole-number zeroes.
• Stopping too early. After one factor of a cubic, factorise the quadratic that is left.

Check yourself

1. Factorise $x^2 - 11x + 28$.

2. Use an identity to find $52 \times 48$.

3. Factorise $8x^3 - 1$.

4. If $x + y + z = 0$, what is $x^3 + y^3 + z^3 - 3xyz$?